Velocity control of hyperbolic partial differential equation systems with single characteristic variable

Abstract

This work addresses the problem of controlling a flow system described by a set of first-order partial differential equations with a single characteristic variable. The manipulated input variable is the characteristic flow velocity of the system while the controlled output is any function of the state variables in the outlet stream. Extending ideas and concepts from geometric control theory for ordinary differential equation systems, the notion of input/output linearization is used as a basis for the controller design. The method of characteristics is used to establish properties of the system dynamics. Because of the presence of a deadtime, the nonlinear system behaves in a manner analogous to linear systems having quasirational transfer functions. It is shown that the zero dynamics is marginally stable. Two types of controller designs are considered. Because of the nature of the zero dynamics, a continuous-time controller is shown to induce oscillations, making it unsuitable for practical use. A novel alternative controller is proposed that will induce a discrete-time linear input/output response in the closed loop. The discrete control action is taken at variable intervals of time equal to the residence time of the fluid entering the system at the start of that interval. The control methodology is implemented on a nonisothermal plug flow reactor and its performance and robustness are evaluated through simulations.